Topics in uniform continuity

Dikr an Dikr anjan Dip artimento di Matematic a e Informatic a, Universit` a di Udine, Via del le Scienze 206, 33 100 Udine, Italy E-mail: dikranja@dimi.uniud.it Du ˇ san R ep ov ˇ s F acu lty of Mathematics and P hysics, and F aculty of Edu c ation, University of Ljubljana, P .O.B. 2964, 1001 Ljub ljana, Slovenia E-mail: dusan.repovs@guest.arnes.si T opics in uniform con tin uit y This pap er collec ts results and op en problems concerning several class es of functions th at generalize u n ifo rm contin uit y in v arious wa y s, includ- ing those metric spaces (generalizing Atsuji spaces) where all con tinuous functions hav e the prop ert y of being close to u nifo rmly conti nuous. Keywords : Closur e op er ator, uniform c ontinuity, A tsuji sp ac e, UA sp ac e, str aight sp ac e, he dgeho g, magic set, lo c al ly c onne cte d sp ac e. Dedicated to the memory of Jan P elant (1950-20 05) 1. Intr oduction The uniform con tin uity of maps b et w een metric or uniform spaces determines a s pecific topic in general top ology . By the end of the fifties and in the seven ties the attent ion was concen trated on those spaces (called UC spaces, or A tsuji spaces) on whic h uniform con- tin uit y coincides with con tin uit y . Of course, compact spaces are UC spaces, but there also exis t non-compact UC spaces (e.g., the uniformly discrete ones). In this survey w e consider several asp ects of uniform con tin uit y and its relationship with con tin uit y . W e start with a discuss ion of the p ossibilit y to capture uniform con tin uit y b y means of the so- called closure op erators [DT]. The most relev ant and motiv ating example of a closure op erator is the usual Kurato wski closure K in the category T op of top ologica l spaces and cont in uous maps. It is c  Dikran Dikr anjan, Du ˇ san Rep ov ˇ s, 2011 2 D. Dikranjan and D. Rep o v ˇ s w ell known that one can describ e the morphisms in T op (i.e., the con tinuou s maps) in an equiv alen t w a y as the maps “compatible" with the Kurato wski closure (see §2.2). In this settin g app ear the uniformly appr o achab le and the we akly uniformly appr o achable (briefly , UA and WUA , resp.) functions (see Definition 1). Section 3 compares the properties UA and WUA with the prop ert y of the u.c. functions whic h ha v e distan t fib ers in an appropriate sense. In Section 4 we consider studies those spaces X on whic h every con tinuou s f uncti on X → R is UA. This class con tains the w ell- kno wn Atsuji spaces, where every con tin uous f unctio n is u.c. [A1, A2, Be, BDC]. Section 5 deals with those metric s paces in whic h the uniform quasi-components of ev ery closed subspace are closed. Ev ery UA space is thin, but there exist complete thin spaces that are not UA. The main result of this section is a separation prop ert y of the complete thin spaces. The last section is dedicated to “additivit y", whic h turns out to b e q uite a non-trivial question in the case of uniform con tin uit y . More precisely , we discuss here the str aight spaces: these are the metric spaces on whic h a con tin uous function X → R is uniformly con tinuou s whenev er the restrictions f ↾ F 1 and f ↾ F 2 on eac h mem- b er of an arbitrary closed binary co v er X = F 1 ∪ F 2 are uniformly con tinuou s. A t the end of the pap er w e collect most of op en problems and question (although some of them can b e found in the main text). W e did not include in this pap er several related issues. One of them, m agic sets , is a topic that app eared in connection with UA functions, even though it has no apparen t connection to uniform con tinuit y . The reader can see [B2, BC1, BC2, BeDi1, BeDi2, CS] for more on this topic. W e are dedicating this surv ey to the memory of the outstandi ng top ologist and our go o d friend Jan P elant , who activ ely w ork ed on this topic and contr ibuted the most relev ant results in this area. T opics in uniform con tin uit y 3 1.1. Notation and termin o l o g y . A top ological space X is called her e ditarily disc onne cte d if all connected componen ts of X are triv- ial, while X is called total ly disc on ne cte d , if all q uasi componen ts of X are trivial [E]. The closure of a subset Y of a top ological space X will be denoted by cl ( Y ) or Y . A ll top ological spaces considered in this pap er are as sumed to b e T yc honoff. A top ological space is said to b e Bair e if it satisfies the Baire Category Theorem, i.e., if ev ery meager subset of X ha s empt y in terior. A Cantor set is a non void zero-dimensional compact metrizable space with no iso- lated p oin ts, i.e., a homeomorphic cop y of the Canto r middle third s set. If X is a top ological space, w e write C ( X ) f or the f amily of all cont in uous real-v alued functions on X , and C n ( X ) for the sets of con tin uous no where constan t real-v alued functions on X . (T o sa y that a cont in uous function f : X → R is no where constan t is equiv alen t to saying that f − 1 ( y ) is no where dense for eac h y ∈ R .) A metr ic space is said to b e uniformly lo c al ly c onn e cte d (shortly , ULC ) [HY, 3-2], if for ev ery ε > 0 there is δ > 0 suc h that any t w o p oin ts at distance < δ lie in a connected set of diameter < ε . In other word s, close p oin ts can b e connected b y small connected sets. F or example, con vex subsets of R m (or any Banac h space) are uniformly lo cally connected. 2. Unif orm continuity vs continuity 2.1. Global view on closure op erators. Closure operators can b e introduced in a quite general con text [DT]. The prominen t examples that inspired this general notion were giv en b y Isb ell (in the category of semigroups, or more generaly , categories of univ ersal algebras) and Salban y (in the category T op of top ological spaces and con tin uous maps). W e briefly recall here the notion of a closure op erator of T op , follo wing [DT, DTW]. A closur e op er ator of T op is a family C = ( c X ) X ∈ T op of maps c X : 2 X − → 2 X suc h that f or ev ery X in T op (i) M ⊆ c X ( M ) for all M ∈ 2 X ; (ii) M ⊆ M ′ ∈ 2 X ⇒ c X ( M ) ⊆ c X ( M ′ ) ; and 4 D. Dikranjan and D. Rep o v ˇ s (iii) f ( c X ( M )) ⊆ c Y ( f ( M )) for all f : X → Y in T op and M ∈ 2 X . A prominen t example is the Kurato wski closure op erator K . Ev ery contin uous f unct ion satisfies the “con tin uity" condition (iii) for every closure op erator C . F or a closure op erator C of T op w e sa y that the set map f : X → Y is C - c onti nuous , if it s atisfies (iii). It is easy to see that a map f : X → Y is con tin uous if and only if it is K - c on tinuous . I n other wor ds, the morphisms in T op can b e detected by a closure op erator (as K - c ontinuous maps). Analogously , a closure op erator of Unif can b e defined as a family C = ( c X ) X ∈ Unif of maps c X : 2 X − → 2 X suc h that for ev ery X in Unif items (i) and (ii) are satisfied, and (iii u ) f ( c X ( M )) ⊆ c Y ( f ( M )) for all f : X → Y in Unif and M ∈ 2 X . W e sa y that C is additive ( idemp otent ) if the equalit y c X ( M ∪ N ) = c X ( M ) ∪ c X ( N ) (resp., c X ( c X ( M )) = c x ( M ) ) alw a ys holds. The Kurato wski closure op erator K is a closure op erator of Unif . Analogously , for a closure op erator C of Unif one can sa y that the set map f : X → Y is C - c ontinuous , if it satisfies (iii u ); f is said to b e total ly c ontinuous if it is C -con tin uous for ev ery closure op erator C of Unif . In the category T op of top ological spaces a map is conti n uous if and only if (iii) is holds for C = K . Hence morphisms in T op are determine d b y the closure op erator K . Can the same b e said of Unif ? This question w as answered in the negativ e in [DP]. W e briefly sk etc h the pro of here. The spaces needed as to ols are the uniformly discrete t w o-p oin t space D = { 0 , 1 } , the one-point compactification N ∞ of the nat- urals N equipped with its unique uniformit y , and t wo uniformly close seq uences, which are not top ologically close X 0 := { ( n, 1 /n ) | n ∈ N } ∪ { ( n, − 1 /n ) | n ∈ N } ⊆ R 2 . Set M a = { ( n, 1 /n ) | n ∈ N } , M b = { ( n, − 1 /n ) | n ∈ N } T opics in uniform con tin uit y 5 and consider the map π : X 0 → D defined by π ( a n ) = 0 and π ( b n ) = 1 for eac h n . Cle arly π is cont in uous but not uniformly con tinuou s, since the op en disjoin t binary co v er X 0 = M a ∪ M b is not uniform. Lemma 1. ([DP]) In the ab ove notation: (a) the map π is C -c on ti nuous for every additive closur e op er- ator C of Unif such that either c ( M a ) \ M a or c ( M b ) \ M a is finite; (b) if c ( M a ) \ M a is infi nite for a closur e op er ator C of Unif , then for ever y metric Bair e sp ac e B ∈ Unif wi thout iso- late d p oints ther e exists a disc onti n uou s map f B : N ∞ → B which is C -c ontin uou s. Theorem 1. ([DP]) L et C b e an additive closur e op er ator of Unif . T hen either π : X 0 → D is C -c ontin uou s or for every m etric Bair e sp ac e wi thout isolate d p oints B ther e exists a disc onti nuous map f B : N ∞ → B which is C -c ontinuous. This sho ws that for every additive closure op erator C of Unif one can find a C -con tin uous map that is not uniformly cotnin uous. Hence, a single closure op erator of Unif cannot detect uniform con tinuit y . This theorem also allo ws us to see whic h C -con tin uous maps fail to b e u.c. It is natural to exp ect that using more than just one closure op erator things ma y c hange. W e shall see now that this is not the case. Ev en a totally con tin uous map, whic h satisfies (iii) for ev ery closure op erator C , is not necessarily uniformly con tin uous. The to ol to achi ev e this result is the follo wing notion in tro duced in [DP] and [BeDi1]: Definition 1. ([BeDi1, DP ]) W e say that f ∈ C ( X, Y ) is UA ( uniform l y approac hable ), if for every c omp act set K ⊆ X and every set M ⊆ X , ther e is a U C function g ∈ C ( X , Y ) whi ch c oincides with f on K and satisfies g ( M ) ⊆ f ( M ) . 6 D. Dikranjan and D. Rep o v ˇ s W e then say that g is a ( K, M ) -approximation of f . If we r e quir e i n the definition of U A that K c onsi sts of a single p oi nt we obtain the we aker notion WUA ( w eakly UA ). Clearly , U A implies W U A . It was s hown in [DP] that R with the natural uniformit y has the prop ert y that every contin uous self map is uniformly approac hable: Example 1. Every f ∈ C ( R ) is UA. In de e d, let K = [ − n, n ] and let M ⊆ R b e an arbitr ary non-empty set. Pick any m 1 ∈ M ∩ ( −∞ , − n ] if this s et is n on-empty, otherwise take m 1 = − n . Cho ose m 2 ∈ M analo gously. Then the function g : R → R define d by g ( x ) =      f ( m 1 ) , if x ≤ m 1 f ( x ) , if m 1 ≤ x ≤ m 2 f ( m 2 ) , if x ≥ m 2 is a ( K, M ) -appr oximation of f . Since “uniformly approa c hable” implies “totally con tin uous” and f ( x ) = x 2 is not uniformly con tin uous it follows that uniform con- tin uit y is not detected eve n b y all closure op erators in Unif . How- ev er, Burk e noticed [BeDi1, Examp le 3.3] that there are con tin uous non- W U A functions on R 2 (in fact, f : R 2 → R , f ( x, y ) = xy , is suc h a function, see Example 5). The next theorem easily follo ws from the definitions: Theorem 2. ([DP ]) Every WUA f unction is total ly c ontinuous. 2.2. Lo cal view on closur e op erators. Ever y additive and idem- p oten t closure op erator of T op or Unif defines a top ology on the underlying set of the space. In this sense, the use of top ologies that mak e certain maps (uniformly) con tin uous in the sequel can b e also view ed as a lo cal use of idemp oten t additiv e closure o p er- ators (i.e., on a single space or an a single pair of spaces, without the axiom (iii)). T opics in uniform con tin uit y 7 2.2.1. T op olo gies τ on R that make a given class of functions F ⊆ R R c oincide with C (( R , τ ) , ( R , τ )) . In the sequel X will b e a met- ric s pace. F ollo wing [C] w e sa y that a class F of functions from X to Y c an b e top olo gize d if there exist top ologies τ 1 on X and τ 2 on Y suc h that F coincides with the class of all con tin uous functions from ( X, τ 1 ) to ( Y , τ 2 ) . The pap er [C] giv es conditions whic h un- der GCH (generalized con tin uum hypothesis) imply that F can b e top ologiz ed. In particular, it is sho wn that (ass umin g GCH) there exists a connected Hausdorff top ology τ on the real line suc h that the class of all contin uous functions in τ coincides with the class of all linear functions. A similar theorem is v alid for the class of all p olynomials, all analytic functions, and all harmonic f unction s. On the other hand, the classes of deriv ativ es, C ∞ , differen tiable, or Darb oux functions cannot b e top ologized. 2.2.2. Char acterization of uniform c ontinuity as a simple c onti nu- ity w.r. t. appr opriate top olo gies. W e recall here the work of Burk e [B1] on char acterization of uniform con tin uit y as a s impl e co n- tin uit y w.r.t. appropriate top ologies (or lo cally defined closure op erators in the abov e sense). Within this setting, the problem b ecomes the question of determining whic h metric spaces X and Y are such that the uniformly con tin uous maps f : X → Y are precisely the con tin uous maps b etw een ( X , τ 1 ) and ( Y , τ 2 ) for some new top ologie s τ 1 and τ 2 on X and Y , resp ectiv ely . Theorem 3. Ther e exist a c on n e cte d close d subset X of the plane, a home omorphism h : X → X , and a c onne cte d Polish top olo gy τ on X such that the c ontinuous self-maps of X ar e pr e cisely the maps h n ( n ∈ Z ) an d the c onstant maps, while the c onti n uou s self- maps of ( X , τ ) as wel l as the unif orm ly c ontinuous self-maps of X ar e pr e cisely the maps h n ( n ≤ 0) and the c onstant maps. 8 D. Dikranjan and D. Rep o v ˇ s 3. Functions with dist ant fibers and unif orm continuity Definition 2. W e say that f ∈ C ( X ) has distan t fib ers (briefly, DF ) if any two distinct fib ers f − 1 ( x ) , f − 1 ( y ) of f ar e at some p ositive distanc e. It is curious to note that this property generalizes t w o antip o dal prop ertie s of a function: • f is constan t ( f has one big fib er) • f has small fib ers (e.g., one-to-one functions, or more gen- erally , functions with compact fib ers, or briefly , KF). 3.1. Uniform con tinuit y coincides with DF for b ounded functions f : R m → R . It is easy to see that U C implies D F f or an y function f : X → R . I ndeed , if d ( f − 1 ( u ) , f − 1 ( v )) = 0 for s ome u 6 = v in R , then an y pair of sequences x n , y n suc h that f ( x n ) = u , f ( y n ) = v and lim n d ( x n , y n ) = 0 witness non-uniform con tin uity of f . Let us verify that D F implies U C f or b ounde d functions f : R m → [0 , 1] . I ndee d, assume that the sequences x n , y n imply non uniform contin uit y of f with d ( x n , y n ) → 0 and | f ( x n ) − f ( y n ) | ≥ ε for ever y n . Then b oundedness of f y ields that f ( x n ) , f ( y n ) can without loss of generalit y b e assumed conv ergen t, i. e., ( x n ) → a, f ( y n ) → b for some a 6 = b in [0 , 1] . L et I n b e the segmen t in R m joining x n and y n . L et a < b and tak e u, v ∈ [0 , 1] with a < v < u < b . Then f ( x n ) ∈ [0 , v ) and f ( y n ) ∈ ( u, 1] for sufficien tly large n since f ( I n ) is an in terv al (being a connected set) con taining f ( x n ) and f ( y n ) . Then u, v ∈ f ( I n ) for sufficien tly large n . Since d ( x n , y n ) → 0 , w e conclude that d ( f − 1 ( u ) , f − 1 ( v )) = 0 , a con tradiction. In the argumen t ab o v e R m can b e replaced b y an y space that is uniformly lo cally connected. This condition cannot b e om itted , since the argumen t function on the circle min us a p oin t is DF but T opics in uniform con tin uit y 9 not UC (indeed, the circle min us a p oin t is not uniformly lo cally connected with resp ect to the metric induced b y the plane). Theorem 4. ([ BDP1, Theorem 3.7]) A b ounde d f unction f ∈ C ( X ) on a unif orm ly lo c al ly c onne cte d sp ac e X is u.c. if and only if it is D F . Boundedness was essen tial to pro v e that D F implies uniform con tinuit y . Indeed, unbounded cont in uous functions R → R need not b e u.c. ev en when they are fi n ite-to-o ne (e.g., x 7→ x 2 ) or eve n inje ctive (e.g., x 7→ x 3 ). The next theorem sa ys that uniform conti n uit y of a b ounded function f ∈ C ( X ) is a prop ert y of its fib ers. In this form the theorem p ermits one to remo ve the h yp othe sis “uniformly lo cally connected". Theorem 5. ([BDP1, Theorem 3.10]) L et ( X, d ) b e a c onne cte d and lo c al ly c onne cte d metric sp ac e. Supp ose that f , g ∈ C ( X , [0 , 1]) have the same family of fib ers and that f is u.c. T hen g is also u.c. Nev ertheless, b oundedn ess cannot b e remo ved, as the pair of functions x 7→ x, x 7→ x 3 on R sho w. It is not clear what is the precise prop ert y of the fibres of f ∈ C ( X, [0 , 1]) whic h gives uniform cont in uit y . Theorem 4 sho ws that it is precisely DF when the space X is uniformly lo cally connected. Ho w to r emo ve b oundedness In order to remo v e b oundedness w e no w consider a generaliza- tion of UC which coincides with UC for b ounded f unct ions. W e start with a notion which is stronger than DF. A function f is said to b e prop er (briefly , P) if the f -preimage of an y compact set is compact. Equiv alen tly , f is a closed map with KF. Even though in general the implication P → K F cannot b e inv erted (e.g., x 7→ arctan x in R ), one can pro v e that P = K F f or un- b ounde d functions R m → R , m > 1 . In particular, this holds for p olynomial functions R m → R . 10 D. Dikranjan and D. Rep o v ˇ s The prop ert y K F implies D F , but it is mu c h stronger. Indeed, U C need not imply K F . This is wh y w e in tro duce the auxiliary notion AP that presen ts a w eak ening of b oth the notion of prop er function and that of UC function . W e plan to show that D F = AP for functions f ∈ C ( X ) on a uniformly lo cally connected space X . Definition 3. ([BDP1]) f ∈ C ( X, Y ) is said to b e AP ( almost prop er ) if f is u.c. on the f -pr eimage of every c omp act set. Ob viously , UC implies AP , whereas b ounded A P f uncti ons are UC. The same argumen t given ab o v e to pro v e U C → D F also pro ves that AP implies D F . On the other hand, with the pro of of Theorem 4 outlined ab o v e one can also show: Lemma 2. ([BDP1, Lemma 3.5]) D F → AP for functions f ∈ C ( X ) on a uniformly lo c al ly c onne cte d s p ac e X . Henc e, AP c oi n - cides w i th D F on uniformly lo c al ly c onne cte d sp ac es. In this w ay w e ha v e ac hiev ed our goal b y replacing U C with AP . Next w e discuss an alternativ e s olution, based on a differen t idea of ch o osing instead of AP a class of functions close to UC in the sense of approxim ation, namely U A (and W U A ). Theorem 6. ([ BDP1, Theorem 3.15]) DF implies UA in uni- formly lo c al ly c onn e cte d sp ac es. The pro of is based on the notion of truncation , which w as implicit in Example 1. Now w e define a differen t kind of truncation (for the general definition see Definition 6). F or a function f : X → R and real n um b ers a ≤ b define the ( a, b ) - trun cat ion as follo ws: f ( a,b ) ( x ) =      f ( x ) , if f ( x ) ∈ [ a, b ] a, if f ( x ) ≤ a b if f ( x ) ≥ b . One prov es that f ∈ D F implies f ( a,b ) ∈ D F , hence f ( a,b ) ∈ U C since it is b ound ed. Now, if K is a compact set and a, b ∈ T opics in uniform con tin uit y 11 R are chosen suc h that f ( K ) ⊆ [ a, b ] , then f ( a,b ) is a ( K, M ) - appro ximation of f if one tak es additional care ab out g ( M ) ⊆ f ( M ) as in Example 1. The ab o v e implication cannot b e in v erted, as the next example sho ws. Example 2. The function f ( x ) = sin x 2 is n ot DF, sin c e any two fib ers of f ar e at distanc e 0. Neve rtheless, ac c or ding to Example 1, f ( x ) is UA. This suggests that the condition DF is to o strong. W e shall consider an appropriat e wea k er v ersion b elo w. Remark 1. Note that the ( a, b ) -trunc ation is differ ent fr om the trunc ation g define d in Example 1 in the c ase X = R . It c an e asily b e shown that if f ([ m 1 , m 2 ]) = [ a, b ] , then g is a trunc ation of f ( a,b ) . 3.2. Distan t conn ect ed comp onents of fib ers. In order to in- v ert the implicati on in Theorem 6 w e need a w eak er form of DF. T o this end we take a closer lo ok at the c onne cte dness structure of the fib ers. First of all w e recall a notion for con tinuous maps in T op . Definition 4. A c ontinuous maps f : X → Y b etwe en top olo gic al sp ac es is c al le d (a) monotone if al l fi b ers of f ar e c onne cte d; (b) ligh t if al l fi b ers of f ar e total ly disc onne cte d. The term in item (a) w as motiv ated b y the fact that f or maps R → R one obtains the usual monotone maps. It is kno wn that ev ery con tin uous map f : X → Y b et we en top ological spaces can b e factorized as f = l ◦ m , where m : X → Z is monotone and l : Z → Y is light . (Notice that this factorization for the constan t map f : X → Y = { y } pro vides as Z exactly the s pace of connected comp onents of X and m : X → Z is the quotien t map ha ving as fib ers the connected comp onen ts of X .) 12 D. Dikranjan and D. Rep o v ˇ s Definition 5. W e say that f has “distan t connected comp o- nen ts of fib ers” ( D CF ) in the sense that any two c omp onents of distinct fib ers ar e at p ositive distanc e. Example 3. L et X b e a metric sp ac e and f : X → R a c ont i nuous map. (a) if f i s monotone, then f is DCF i f and only i f f i s DF. (b) If f i s light, then i t is DCF . In p articular, the f unction f : (0 , 1 /π ] → R define d by f ( x ) = sin 1 /x , as wel l as the function fr om Example 2, ar e DC F ( b eing light), but any two non-empty fib ers of f ar e at distanc e 0. Theorem 7. ([BDP1, Theorem 4.3]) UA i mplies DC F f or f ∈ C ( X ) and arbitr ary metric sp ac es X . In fact, if C a and C b are t w o connected comp onen ts of fibres of f at distance 0, then for K = { a, b } and M = C a ∪ C b the function f has no ( K, M ) -appro ximation. Along with Example 1 this give s: Corollary 1. Every c ontin uou s r e al-value d function R → R is DCF. A ctually , w e shall see b elo w that even W U A implies D C F for f ∈ C ( R m ) (see Theorem 8), therefore, D F → U A → W U A → D C F for f ∈ C ( R m ) . Hence all they coincide for p olynomial functions (or f unctions with finitely man y connected comp onen ts of fibres). Let us put all these implications in the follo wing diagram, where the equi v alence (1) for ULS spaces is given b y Lemma 2 and the implication (2) for ULS spaces is giv en by Theorem 6. The implication (3) for R n follo ws from these t w o implications and the trivial implication U A → W U A . The implication (4) will b e pro v ed in Theorem 8 b elo w which gives a m uc h stronger result. T opics in uniform con tin uit y 13 ✻ ❄ ✻ ✲ ✲ ✲ KF ✲ ❄ ✻ UA (2) (3) (4) WUA AP DF UC (1) DCF Diagram 1 W e shall see b elo w that (4) is not an equiv alence. Th is moti- v ated the in tro duction of the follow ing weak er version of U A in [CD2]: a function f : X → R is said to b e U A d ( densely uniformly appr o achable ) if it admits uniform h K, M i -appro x imati ons for ev- ery dense s et M and f or every compact set K . Analogously , one can define W U A d . Theorem 8. W U A d c oincides with D C F for f ∈ C ( R m ) . The pro of requires a new form of w eak UC based on truncations: Definition 6. g ∈ C ( X ) is a truncation of f ∈ C ( X ) if the sp ac e X c an b e p arti tione d in two p arts X = A ∪ B so that g = f on A and g is c onstant on e ach c onne cte d c om p onent of B (that is, g must b e c onstant on e ach c onne cte d c omp onent of { x ∈ X : f ( x ) 6 = g ( x ) } ). This motiv ates the int ro duction of the class T U A of trunc ation- U A f unct ions, that is, functions f ∈ C ( X ) such that for ev ery compact set K ⊆ X there is a u.c. truncation g of f whic h coin- cides with f on K . The follow ing is easy to pro v e: 14 D. Dikranjan and D. Rep o v ˇ s Theorem 9. ([BDP1]) TUA im pli es DCF on every lo c al ly c on- ne cte d sp ac e. Indeed, if C a and C b are tw o connected comp onen ts of fibres of f at distance 0, then for K = { a, b } the function f has no UC K -truncations. The pro of of Theorem 8 splits in three steps (see in Diagram 2 b elo w) • Step 1: ([BDP1, Corollary 7.4]) D C F → T U A for f ∈ C ( R m ) • Step 2 : ([CD2, Th eorem 3.1]) T U A → U A d for f ∈ C ( R m ) • Step 3: ([BDP1], [CD2, Corollary 4.2]) W U A d → D C F for f ∈ C ( R m ) . Step 1 and Theorem 9 ensure the equiv alence (4) for f ∈ C ( R m ) in Diagram 2. Step 2, the trivial implication U A d → W U A d , Step 3 and the equiv alence (4) imply (5). This pro v es all f our equiv alences for f ∈ C ( R m ) in the righ t square of Diagram 2. The remaining three implications (1), (2) and (3) are trivial. ✻ ✲ ✲ ✲ ✲ ❄ ✻ ✛ ✛ ✻ ❄ UA d (2) (3) (5) WUA d WUA UA (1) (4) DCF TUA Diagram 2 In view of the f our equiv alences in the righ t square of Diagram 2, the next example shows that the implications (2) and (3) cannot b e in v erted. T opics in uniform con tin uit y 15 Example 4. ([CD2 , §5]) In C ( R 2 ) , T UA do es not imply WUA. It remains unclear whether the remaining last implication (1) of Diagram 2 can b e inv erted for f ∈ C ( R m ) (see Problem 1). 4. U A sp a ces The main ob jectiv e of this section are the UA spaces – spaces where ever y con tin uous function is UA. The first example of th is kind is R (Example 1). The motiv ation to in troduce these spaces are the well k no wn Atsuji spaces. Here w e recal l some results from [BeDi1] and w e an ticipate some of the pricipal results from [BDP3] whic h giv e further motiv ation for s tudying UA functions. The next definition will b e used in the sequel. Definition 7. Two subsets A, B of a top olo gic al sp ac e X ar e said to b e separated if the closur e of e ach of them do es not me et the other (this i s e quivalent to sayin g that A and B ar e clop en in A ∪ B ). So X is c onne cte d i f and only if it c annot b e p artitione d in two sep ar ate d sets. A s ubset S of X separates the nonempty sets A and B if the c omplement of S c an b e p artitione d in two sep ar ate d sets, one of which c ontain s A , the other c ontains B (se e [K , §16, VI ] ). 4.1. UA spaces. Sev eral criteria for UA-ness are given, among them the follo wing lo oks most s pectacular: Theorem 10. ([BDP3]) L et X b e a U A sp ac e and let A, B b e disjoint close d unif orm ly c onne cte d subsets of X . T hen ther e is a c ol le ction { H n | n ∈ N } of nonempty close d subsets of X such that for every n , (1) H n +1 ⊆ H n ; (2) H n sep ar ates A and B ; and (3) H n is c ontain e d in a finite union of b al ls of diameter < 1 /n . 16 D. Dikranjan and D. Rep o v ˇ s A ctually , this prop ert y can b e pro v ed for a larger class of spaces discussed in §6, where a relev an t prop ert y is obtained in the case when X is complete (Theorem 21). The f ollo wing construction whic h pro duces UA spaces from trees and compact sets placed at their vertice s w as giv en in [BDP1 ]. Definition 8. A metric sp ac e X is a tree of compact s ets { K n : n ∈ ω } if X = S n ∈ ω K n wher e e ach K n is c omp act and | K n +1 ∩ S i ≤ n K i | = 1 . (a) Given a subset I ⊆ ω , we say that the subsp ac e X I = S n ∈ I K n of X = S i ∈ ω K i is a subtree of X if for every n, m with n < m , i f n ∈ I and K n ∩ K m 6 = ∅ , then m ∈ I . (b) A tr e e of c om p act sets X = S i ∈ ω K i is said to b e tame if every K i has an op en neighb ourho o d whi ch interse cts only finitely m any K j ’s and every two disj oint subtr e es of X ar e at a distanc e > 0 . It is easy to see that the circle min us a p oin t can b e represen ted as a tree of compact sets, but none of these trees is tame. The next theorem shows the reason f or that (the circle min us a p oin t is not a U A s pace) . Theorem 11. [BDP3] If X = S i ∈ ω K i is a tame tr e e of c omp act sets { K n : n ∈ ω } , then X is U A . Examples of tame trees are giv en in Figure 1 (see ladders B and C). 4.2. Non-UA spaces: Hedghogs and some necessary condi- tions. Let α b e a cardinal. In the sequel H α denotes the he dgho g with α spikes (see [E, Example 4.1.5], note that H a is separable if and only if a = ω ). Recall the definition of the cardinal b as the minimal cardinalit y of an unbounded f amily of functions f : ω → ω with resp ect to the partial pr e ord er f ≤ ∗ g if f ( n ) ≤ g ( n ) for all but finite n um b er n ∈ ω (see [vD]). In ZFC ω 1 ≤ b ≤ 2 ω , and b = 2 ω consisten tly (for example under MA or CH, see [vD] for more detail). T opics in uniform con tin uit y 17 Surprisingly , one has the follo wing indep endenc y result: Z F C cannot decide whether the smallest non-separable hedghog H ω 1 is UA. More precisely , the follo wing holds: Theorem 12 . [BDP3] L et α b e a c ar din al. If α < b then H α is UA, w her e as if α ≥ b then H α is not even W U A . In particular , under CH the space H ω 1 is not UA, while in mo d- els of ZF C where ¬ C H & M A holds, one has ω 1 < b , so H ω 1 is U A . W e s how b elo w that ev ery space H α is T U A (Corollary 2). Hence T U A 6→ W U A for nonseparable spaces. W e also consider the follo wing space whic h is more general than the hedghog H α of α spik es: Definition 9. L et α b e an i nfinite c ar di n al. A metric sp ac e ( X , d ) is c al le d a hedghog of compact sets { K λ : λ ∈ α } if X = S λ ∈ α K λ wher e e ach K λ is c omp act with mor e than one p oint and ther e exists p ∈ X such that (1) K λ ∩ K λ ′ = { p } f or λ 6 = λ ′ ; an d (2) ∀ x, y ∈ X [ d ( x, y ) < max { d ( x, p ) , d ( y , p ) } → ∃ λ < α [ x, y ∈ K λ ]] . Sometimes we pr efer to say m or e pr e cisely: a hedghog of α com- pact sets . Definition 10. L et X b e a uniform s p ac e, let f ∈ C ( X ) and let K b e a c omp act subset of the sp ac e X . The minimal K -trunc ation f K of f is define d as the (inf K f , sup K f ) -trunc ation of f . Theorem 13. [BDP3 ] L et X b e a he dgho g of c omp act sets and K b e a c omp act subset of X c ontaining p . Then for every c ontinuous function f ∈ C ( X ) the (inf K f , sup K f ) -trunc ation of f is u.c. This gives the follo wing: Corollary 2. Every he dgho g of c omp act sets is T U A . Another source of UA spaces is giv en b y the f ollowin g: 18 D. Dikranjan and D. Rep o v ˇ s Theorem 14. [BDP3 ] Every uniformly zer o-dimensional sp ac e is UA. In [BDP3] the Can tor s et is ch aracterized as the only compact metrizable space M s uc h that eac h subspace of M is UA. Theorem 15. [BDP3] The only manifolds which ar e WUA ar e the c omp act ones and the r e al line. No w we see that a metric space havin g a contin uous function whic h is not uniformly con tin uous, necessarily also has a b ounded uniformly approac hable function that is not uniformly con tin uous. Hence, in some s ense, UA is “closer to con tin uit y than to uniform con tinuit y". Theorem 16. [BDP3] A metric sp ac e X is U C if and only if every b ounde d uniformly appr o achable function is uniformly c ontinuous, so that a sp ac e X with C u ( X ) = C ua ( X ) is ne c essarily a U C sp ac e. In order the get a necessary condition f or b eing a WUA space, the follo wing notion was prop osed in [BeDi1]: Definition 11. L et X b e a uniform sp ac e. A fami ly of pseudo- h yperb olas in X is given by a c ountab le family { H n } of disjoint subsets of X such that f or every n ∈ N : (1) H n is close d and uniformly c onne cte d; (2) H n ∪ H n +1 is uniformly c onne cte d; (3) H n ∩ S m>n H n = ∅ ; and (4) the set H = S n H n is not close d i n X . This notion w as inspired b y the follo wing example due to Burke. Example 5. A fami ly of pseudo-hyp erb olas in R 2 is given by the sets H n = { ( x, y ) : ( xy ) − 1 = n } . Theorem 17. ([BeDi1]) I f a normal uniform sp ac e X has a f amily of pseudo-hyp erb olas, then X is not W U A . Theorem 18. ([BeDi1]) L et X b e a sep ar able uniform sp ac e and supp ose that ther e exists f ∈ C ( X ) with c ountable fib ers without T opics in uniform con tin uit y 19 non-c onstant uniformly c onti nuous trunc ations. Then X is n ot W U A . Example 6. Now we give thr e e non- W U A examples of subsets X 1 , X 2 and X 3 of R 2 that c ontain no pseudo-hyp erb olas. T o pr ove that they ar e n ot W U A one c an apply T he or em 18. So it is n e c- essary to find, in e ach c ase, a c onti nuous function with c ountable fib ers and without non-c onstant uni formly c ontin uou s trunc ati ons. (a) The sp ac e X 1 is the unit cir cle mi nus a n on-empty fini te set. So X 1 c an b e identifie d with a c ofinite subset of the set of c omplex numb ers e iθ with 0 < θ < 2 π . Define f : X 1 → R by f ( e iθ ) = θ . Then f i s non- W U A . It i s e asy to se e that X 1 c ontains no pseudo-hyp erb olas. (b) The sp ac e X 2 c onsists of the union of the two hyp erb o- las H 1 = { ( x, y ∈ R 2 : x ≥ 0 , y ≥ 0 , xy = 1 } an d H 2 = { ( x, y ∈ R 2 : x ≥ 0 , y ≥ 0 , xy = 2 } . Obviously, X 2 c ontains no pseudo-hyp erb olas. Define f : X 2 → R as f ol- lows. If ( x, y ) ∈ H 1 , then set f ( x, y ) = e x . If ( x, y ) ∈ H 2 , then set f ( x, y ) = − e − x . It is e asy to se e that thi s works. (c) L et X 3 b e the sp ac e fr om Diagr am 3. It c ontains no pseudo- hyp erb olas. Define f : X 3 → R by identifying X 3 with the subsp ac e of R 2 c onsisting of the union of the two verti- c al axes x = − 1 and x = 1 , to gethe r with the horizontal se gments I n = { ( x, n ) ∈ R 2 : − 1 ≤ x ≤ 1 } ( n ∈ N ). L et f ( − 1 , y ) = − y , f (1 , y ) = y . T his defines f on the two axes of the ladder. On e ach horizontal se gment I n , f i s line ar. This uniquely defines f sinc e we have alr e ady define d f on the extr ema of the horizontal se gments I n . f is pseudo- monotone, so e ach trunc ation of f is an ( a, b ) -trunc ation. Any such non-c onstant trunc ation is not uniformly c ontin- uous. Sinc e f has c ountable fib ers, The or em 18 applies and thus X 3 is not W U A . 20 D. Dikranjan and D. Rep o v ˇ s Diagram 3: A s ubset of R 2 5. Thin sp a ces The class of top ologica l spaces X ha ving connected quasi com- p onen ts is closed under homotop y t yp e and it con tains all compact Hausdorff spaces (see [E , Theorem 6.1.23]) and ev ery subset of the real line. Some sufficien t conditions are given in [GN] (in ter ms of existence of Vietoris con tinuou s selections) and [CMP] (in terms of the quotien t space ∆ X in whic h eac h quasi-comp onen t is iden ti- fied to a p oin t), but an easily-stated description of this class do es not seem to b e a v ailable (see [CMP]). The situation is compli- cated even in the case when all connected comp onen ts of X are trivial, i.e., when X is hereditarily disconnected. I n these terms the q uestion is to distinguish b et ween hereditarily disconnected and totally disconnected spaces (examples to this eff ect go back to Knaster and Kurato wski [KK]). The connectedn ess of the quasi comp onen t (i.e., the coincidence of the quasi comp onen t and the connected comp onen t) in top o- logical groups is also a rather hard question. Although a lo cally compact s pace do es not need to ha v e connected quasi components T opics in uniform con tin uit y 21 [E, Example 6.1.24], all lo cally compact groups ha v e this prop ert y . This is an easy consequence of the w ell kno wn fact that the con- nected comp onen t of a lo cally compact group coincides with the in tersectio n of all op en subgroups of the group [HR, Theorem 7.8]. All counta bly compact groups w ere shown to hav e this prop ert y , to o ([D3 ], see also [D2 , D4 ]). Man y examples of pseudo compa ct group where this propert y s trong ly f ails in differen t asp ects, as w ell as further information on quasi comp onen ts in top ological groups, can b e found in ([D1, D2, D4], see also [U] for a planar group with non-connect ed quasi comp onen ts). Let us recall the definition of the quasi c omp onent Q x ( X ) of a p oin t x in a top ological space X . This is the set of all p oin ts y ∈ X such that f ( y ) = f ( x ) for every con tin uous function f : X → { 0 , 1 } , where the doubleton { 0 , 1 } is discrete. Analogously , giv en a uniform space X and a p oin t x ∈ X the uniform quasi c omp onent of x consists of all p oin ts y ∈ X suc h that f ( y ) = f ( x ) for ev ery uniformly contin uous function f : X → { 0 , 1 } , where the doubleton D = { 0 , 1 } has the uniformly discrete structure (i.e., the diagonal of D × D is an en tourage) . W e denote b y Q u x ( X ) the uniform quasi comp onen t of x . In these terms we ha v e the follo wing inclusions C x ( X ) ⊆ Q x ( X ) ⊆ Q u x ( X ) , ( ∗ ) where C x ( X ) denotes the connected comp onen t of x . No w w e can in troduce the relev an t notion for this section: Definition 12. A uniform sp ac e X is sai d to b e thin if f or every close d subset Y of X and every y ∈ Y , the uniform quasi c omp o- nent of y in Y is c onne cte d. It is easy to see that all inclusions in (*) b ecome equalities in the case of compact spaces. Hence compact spaces are thin. Th is also f ollo ws from the more general prop ert y given in Theorem 20. Definition 13. F or thr e e subsets A, B and S of a top olo gic al sp ac e X we say that S cuts b etwe en A and B if S interse cts every 22 D. Dikranjan and D. Rep o v ˇ s c onne cte d set which me ets b oth A and B . (If S i s empty this me ans that ther e is no c onne cte d set which me ets b oth A and B .) If a set separates A and B (see Definition 12), then it also cuts b et wee n A and B , but the con v erse is false in general. Definition 14. W e say that a unif orm sp ac e X has the compact separation p r op ert y (briefly CSP ), if f or any two disjoint close d c onne cte d subsp ac es A an d B ther e is a c omp act set K dis j oint fr om A and B such that every neighb ourho o d of K disjoint f r om A and B s ep ar ates A and B (c on s e quently K interse cts every close d c onne cte d set which me ets b oth A an d B , se e Definition 7). It is easy to see that eve ry compact space has CSP since disjoin t compact sets are alwa ys s epara ted. It wa s pro v ed in [BDP3, Lemma 3.2] that if a metric space X con tains t w o disjoint closed sets H and K and a p oin t a ∈ H s uch that the uniform quasi comp onen t of a in H ∪ K in tersects K , then X is neither thin nor U A . This y ields the followin g corollary: Corollary 3. Two disj oi nt close d unif ormly c onn e cte d subsets A, B of a thi n metric sp ac e X ar e at p ositive distanc e. The follo wing notion is relev an t to the description of thin s paces. Definition 15. Given two distinct p oints a, b of a metric sp ac e X such t hat the uniform ly c onne cte d c omp onent of a c ontains b , ther e exists for e ach n a finite set L n ⊂ X whose p oints form a 1 /n -chain f r om a to b . W e say that the sets L n , to gethe r with a and b , form a ( di scr ete ) garland , if ther e is an op en subset V of X which sep ar ates a and b and such that V ∩ S n L n is close d (and discr ete). One can giv e a c haracterizati on of the thin of metric spaces in terms of existence of garlands in the space. Prop osition 1. [BDP3] F or a metric sp ac e X the fol lowing c onditions ar e e quivalent: (a) X is thin ; T opics in uniform con tin uit y 23 (b) X c ontains no garlands; (c) X c ontains no discr ete garlands. The main result of the pap er [BDP3 ] is the followin g: Theorem 19. Every c omplete thin metric sp ac e has CSP . A large source of thin spaces is pro vided b y US spaces. Theorem 20. ([BDP3 ]) Every UA metric sp ac e sp ac e i s thin. Theorem 19 f ollows from the follo wing more precise result: Theorem 21. L et X b e a c omplete thin metric sp ac e and let A, B b e disjoint close d c on ne cte d subsets of X . T hen: (1) ther e is a c omp act set K such that e ach neighb ourho o d of K disjoi nt f r om A ∪ B s ep ar ates A and B ; (2) henc e K i n terse cts every close d c on n e cte d set which me ets A and B ; (3) if X is also lo c al ly c omp act, ther e is a c omp act set K ′ which sep ar ates A and B . 5.1. CSP vs thin and compl ete. The next ex amples show that the implications in Theorems 19 and 20 cannot b e inv erted. Example 7. Ther e exist many examples of sep ar able metric sp ac e with C SP w hi ch ar e not thin: (i) the cir cle mi nus a p oint (it has two close d c onne cte d subsets at distan c e zer o, so it c annot b e thin by C or ol lary 3); (ii) the r ationals Q (uniform ly c onne cte d non-c onne cte d, henc e not thi n). None of the ab o v e examples is complete. Here is an example of a complete s epara ble metric s pace with CSP whic h is not thin. Example 8. L et H 1 and H 2 b e the br an che s of hyp erb olas { ( x, y ) ∈ R 2 : xy = 1 } and { ( x, y ) ∈ R 2 : xy = 2 } , r esp e ctively, c ontaine d in the first quadr ant. Then the sp ac e X = H 1 ∪ H 2 with the metric in- duc e d fr om R 2 is a c omplete sep ar able sp ac e. Si nc e H 1 and H 2 ar e 24 D. Dikranjan and D. Rep o v ˇ s c onne cte d and at distanc e zer o, it f ol lows fr om C or ol lary 3 that X is not thin. On the other hand, the empty set sep ar ates the close d c onne cte d sets H 1 and H 2 . So if A and B ar e close d c onne cte d disjoint sets in X , it r emains to c onsider only the c ase when b oth A and B ar e c ontaine d in the same c omp onent H i ( i = 1 , 2 ). Now A and B c an b e sep ar ate d by a p oint. Theorem 19 can b e giv en the f ollo wing more general f orm. A metrizable space X with compatible metrics d 1 , d 2 suc h that ( X, d 1 ) is complete (i.e. X is ˇ Cec h-complete) and ( X , d 2 ) is thin admits also a compatible metric d suc h that ( X, d ) is complete and thin (namely , d = max { d 1 , d 2 } ). H ence eve ry ˇ Cec h-complete metriz- able space that admits a compatib le thin metric has CSP . This explains why the spaces in (i) ab o v e and Example 8 hav e CSP . Although complete ness was essen tially used in the pro of of The- orem 19 , it is not clear whether it is in fact necessary , in other w ords: Question 1. Ar e ther e examples of thin sp ac es that do n ot have CSP? What ab out U A sp ac es? As the follo wing example s ho ws, neither thinness nor U A -ness is preserve d by passing to completions, th us an immediate appli- cation of Theorem 19 (via passage to completions) cannot help attempts to answ er Question 1. Example 9. T her e is a U A m etric sp ac e whose c ompletion is not thin (henc e n ot U A ). L et X = S n ∈ N { 1 /n } × I , wher e I is the unit interval [0 , 1] ⊂ R , let a = (0 , 0) , b = (0 , 1) and Y = X ∪ { a, b } . W e put on Y the fol lowing m etric. The distanc e b etwe en two p oints ( x 1 , y 1 ) and ( x 2 , y 2 ) is | y 1 − y 2 | if x 1 = x 2 . Otherwise t he distanc e is the minimum b etwe en y 1 + y 2 + | x 1 − x 2 | and (1 − y 1 ) + (1 − y 2 ) + | x 1 − x 2 | . With this metric Y is the c ompletion of X and the two p oints a, b ar e the limits for n → ∞ of (1 /n, 0) and (1 /n, 1) , r esp e ctively. The sp ac e Y is not thin sin c e ther e is a garland c onsistin g of a , b and h L n | n ∈ N i wher e L n is a 1 /n -chain b etwe en a and b in { 1 /n } × I . The sp ac e X is U A s inc e X is a union of a chain of T opics in uniform con tin uit y 25 c omp act sets, e ach attache d to the next by at most on e p oint (se e [BeDi1, Theorem 11.4] and the intr o duc tion). 5.2. Thin do es not imp l y UA for com plete m et ric spaces. W e give an example of a complete connected thin metric space that is not U A . Example 10 . F or any c ar din al α the he dgeho g J ( α ) i s thin. In- de e d, if J ( α ) wer e not thin , then by Pr op osition 1, i t would c on tain a discr ete garland a, b, h L n | n ∈ N i . L et V b e an op en set sep ar at- ing a, b such that V ∩ S n L n is close d and discr ete . The min i mal c onne cte d set C c ontaining a, b must non-trivial ly i nterse ct V , so it c ontains an op en interval I on one of the spikes. Now, when- ever 1 /n is less than the diameter of I , L n must in terse ct I , so V ∩ S n L n has an ac cumulation p oin t, which is a c ontr adiction. This gives the follo wing immediate corollary of Theorem 12 Corollary 4. F or every α ≥ b the he dgeho g J ( α ) is thin (so has the pr op erty CSP), but n ot UA. The space J ( α ) is not separable for α > ω . On the other hand, b > ω ([vD]), hence the ab o v e examples are not separable. A c- cording to Theorem 12 the hedgehogs J ( α ) are U A for all α < b , so one cannot get in this wa y an ex ample of a separable space with the ab o ve prop erties (see Question 4). 6. Gluing unif orml y continuou s functions It is well- kno wn fact that a map f : X → Y b et we en top ological spaces is con tinu ous whenev er its restriction to eac h mem b er of a lo cally finite closed cov er of X is con tin uous. This s ection is dedicated to the analogue of this prop ert y f or uniform con tin uity . 6.1. Straigh t spaces. In order to c haracteriz e the spaces where uniformly con tinuou s functions can b e glued as the con tin uous ones, the f ollowin g definition was in troduced in [BDP4]: 26 D. Dikranjan and D. Rep o v ˇ s Definition 16. A sp ac e X is c al le d straigh t if whenever X is the union of t w o close d sets, then f ∈ C ( X ) is u.c. if and only if its r estriction to e ach of the close d sets is u.c. Apparen tly , it w ould b e more natural to ask ab out the p ossibilit y to glue toget her finite n um b er of u.c. functions instead of just t w o. The follo wing geometric criterion obtained in [BDP4] justifies this c hoice. T w o subsets A and B of a uniform space X are called U -distant (or simply , distant ) if there exists an ento urage U such that A [ U ] ∩ B = ∅ (or equiv alent ly , there exists an en toura ge U suc h that A ∩ B [ U ] = ∅ ). Definition 17. L et ( X , U ) b e a uniform sp ac e. A p air C + , C − of close d sets of X is said to b e u-placed if C + U and C − U ar e distant for every entour age U , wher e C + U = { x ∈ C + | x 6∈ ( C + ∩ C − )[ U ] } C − U = { x ∈ C − | x 6∈ ( C + ∩ C − )[ U ] } . Remark 2. (a) In the c ase of a metric sp ac e ( X , d ) w e al- ways c onsider the metric uniform i ty of X , s o that in such a c ase a p air C + , C − of close d sets of X is u-placed if d ( C + ε , C − ε ) > 0 holds for every ε > 0 , wher e C + ε = { x ∈ C + : d ( x, C + ∩ C − ) ≥ ε } an d C − ε = { x ∈ C − : d ( x, C + ∩ C − ) ≥ ε } . (b) Note that C + ε = C + and C − ε = C − when C + ∩ C − = ∅ i n Definition 17. Henc e a p artition X = C + ∪ C − of X into clop en sets i s u-plac e d if and only i f C + , C − ar e unif ormly clop en (a subset U of a uniform sp ac e X is uniformly clop en if the char acteristic function X → { 0 , 1 } of U is uniformly c ontinuous wher e { 0 , 1 } is discr ete). Theorem 22. F or a uni form sp ac e ( X, U ) and a p air C + , C − of close d sets the fol lowing statements ar e e quivalent: (1) the p air C + , C − is u-plac e d; (2) a c ontinuous function f : C + ∪ C − → R is u.c. whenever f | C + an d f | C − ar e u.c. T opics in uniform con tin uit y 27 (3) same as (2) with R r eplac e d by a gener al uniform sp ac e ( M , V ) The next theorem extends the defining prop ert y of straight spaces to arbitrary finite pro ducts. Theorem 23. [BDP4] If a metric sp ac e X is str aight and X c an b e wri tten as a union of fini tely many close d sets C 1 , . . . , C n it fol lows that f ∈ C ( X ) is u.c. if and only if e ach r estriction f | C k ( k = 1 , 2 , . . . , n ) of f is u.c. Definition 18. L et X b e a metric sp ac e. W e say that X is W U LC , if for ev ery p air of se quenc es x n , y n in X wi th d ( x n , y n ) → 0 and such that the set { x n } is close d and discr ete 1 ther e exist a n 0 ∈ N and c onne cte d sets I ( x n , y n ) c ontaining x n and y n for every n ≥ n 0 in such a way that the diam I ( x n , y n ) → 0 . Prop osition 2. Every W U LC sp ac e is str aight. Pr o of. Assume X is the union of finitely man y closed sets F 1 , . . . , F m and the restriction of a f unction f ∈ C ( X ) to eac h of the closed sets F k is u.c. W e hav e to c hec k that f is u.c. Pic k ε > 0 and assume that | f ( x n ) − f ( y n ) | ≥ 2 .ε ( ∗ ) for some x n , y n suc h that d ( x n , y n ) → 0 . It is clear, that the sequence x n cannot ha ve an accum ulation p oin t x in X , s ince then some subsequence x n k → x and also y n k → x . N o w the con tin uity of f w ould imply | f ( x n k ) − f ( x ) | → 0 and | f ( y n k ) − f ( x ) | → 0 . Consequen tly , | f ( x n k ) − f ( y n k ) | → 0 con trary to (*). Therefore, the double seq uence x n , y n satisfies the condition (a) of Definition 18. Therefore, for large enough n w e hav e a connected s et I n con taining x n , y n suc h that diam I n < δ . W e can c hoose δ > 0 suc h that also | f ( x ) − f ( y ) | < ε/m whenev er x, y b elong to the same closed s et F k and d ( x, y ) < δ . Note that f ( I n ) is an interv al with length ≥ 2 · ε co v ered b y m subsets f ( I n ∩ F k ) , k = 1 , . . . , m 1 so that also the set { y n } is closed and discrete. 28 D. Dikranjan and D. Rep o v ˇ s eac h with diameter ≤ ε/m . This leads to a contra diction since an in terv al of length ≥ 2 · ε cannot b e cov ered b y m sets of diameter ≤ ε/m .  As a corollary we obtain that ULC spaces are straigh t. Theorem 24. L et ( X , d ) b e lo c al ly c onne cte d. Then ( X , d ) is str aight if and only if it i s uniformly lo c al ly c onne cte d. Theorem 25. L et ( X , d ) b e a total ly disc onn e cte d metric sp ac e. Then X is str aight if and only if X i s UC. 6.2. Stabilit y pr o p erties of straigh t spaces. The next the- orem sho ws that straigh tness sp ectacularly fails to b e preserv ed under taking closed spaces. Theorem 26. F or every m etric sp ac e X wi th I n d X = 0 the fol lowin g ar e e quivalent: (1) X is UC; (2) every close d subsp ac e of X i s str ai ght; (3) whenever X c an b e wri tten as a union of a lo c al ly finite family { C i } i ∈ I of close d sets we have that f ∈ C ( X ) is u.c. if and only if e ach r estriction f | C i of f , i ∈ I , i s u.c. The follo wing notion is relev an t for the description of the dense straigh t subspaces. Definition 19 . ( [BDP5] ) An extension X ⊆ Y of top olo gic al sp ac es is c al le d tigh t if for every close d binary c over X = F + ∪ F − one has (6.1) F + Y ∩ F − Y = F + ∩ F − Y . With this notio n one can chara cterize straigh tness of extensions. Theorem 27. ( [BDP5 ] ) L et X , Y b e metric sp ac es, X ⊆ Y and let X b e dense in Y . T hen X is str aight if and only if Y is st r aight and the extension X ⊆ Y is ti ght. T opics in uniform con tin uit y 29 6.3. Pro ducts of straight spaces. Here w e discuss preserv ation of straigh tness under pro ducts. Nishijima and Y amada [NY] prov ed the follo wing Theorem 28. ([NY]) L et X b e a str aight sp ac e. Then X × K is str aight for e ach c omp act s p ac e K if and only if X × ( ω + 1) is str aight. The next lemma easily follo ws from the definitions. Lemma 3. ([BDP6]) A pr o duct X × Y is ULC if and on ly if b oth X and Y ar e ULC. The next propos ition, pro v ed in [BDP6], plays a crucial role i n the pro of of Theorem 29: Prop osition 3. I f X × Y is str aight, then X i s ULC or Y is pr e c omp act. Theorem 29. ([BDP6]) The pr o duct X × Y of two metric sp ac es is str aight if and only if b oth X and Y ar e str aight and one of the fol lowin g c onditions holds: (a) b oth X and Y ar e pr e c omp act; (b) b oth X and Y ar e ULC; (c) one of the sp ac es is b oth pr e c omp act and ULC. It turns out that the straigh tness of an infinite pro duct of U LC spaces is related to connectedness: Theorem 30 . ([BDP6]) L et X n b e a ULC sp ac e for e ach n ∈ N and X = Π n X n . (a) X is UL C i f and only if al l but finitely many X n ar e c on- ne cte d. (b) The fol lowin g ar e e quivalent: (b 1 ) X is st r aight. (b 2 ) either X is ULC or e ach X n is pr e c om p act. This theorem completel y settles the case of infinite p o wers of ULC space: 30 D. Dikranjan and D. Rep o v ˇ s Corollary 5. L et X b e ULC. Then (a) X ω is ULC if and only i f X is c onne cte d; (b) X ω str aight if and only if X is either c onn e cte d or pr e c om- p act. The ab o v e results lea v e op en the question ab out when infinite pro ducts of precompact straigh t spaces are still straigh t (see Ques- tions 7 and 6). 7. Questions Our first op en problem is ab out the implication (1) in Diagram 2: Problem 1. ([CD2, Problem 1.4]) Do es W U A imply U A in C ( R n ) ? What ab out C ( R 2 ) ? 7.1. Questions on U A functions and UA spaces. A general question is to c haracterize the U A and W U A spaces and functions. W e list b elo w more sp ecific questions ([BeDi1]). (1) Characterize the U A functions f : R 2 → R . (2) Characterize the U A subsets of R . (3) Characterize the topological spaces whic h admit a U A uni- formit y , and those which are U A under every uniformit y compatible with their top ology . Do es the latter class of spaces also include the U C s paces? (4) Do W U A and U A coincide f or connected spaces? (5) Supp ose that a uniform s pace X has a dense U A subspace. Do es it follo w that X is U A ? (This fails for W U A accord- ing to Example 7.7 from [BeDi1].) (6) Let X b e the pushout of t wo W U A spaces ov er a single p oin t. Is X W U A ? (This holds f or U A b y Theorem 11.1 from [BeDi1].) (7) Supp ose that ev ery pseudo-monotone function f ∈ C ( X ) is U A . Is then X a U A s pace ? (8) Define 2 - U A similarly as U A but with the set K of cardi- nalit y at most 2 . Is then 2 - U A equiv alen t to U A ? T opics in uniform con tin uit y 31 7.2. Questions on thin spaces. The next question is related to Theorem 19: Question 2. Is it true that a c omplete thi n uniform sp ac e has C S P ? What ab out a c omplete U A uniform sp ac e? Our next q uestion is ab out ho w m uc h one needs the fact that uniform q uasi comp onen ts are connected. Question 3. Is it true that every c omplete metric s p ac e X such that every close d subsp ac e of X has c onne cte d quasi c om p onents ne c essarily has CSP ? Question 4. Is it true that every (c omplete) metric thin sep ar able sp ac e is U A ? 7.3. Questions on straight s p ace s. Theorem 27 gives a crite- rion for straigh tness of a dense subspace Y of a straigh t space X in terms of prop erties of the em bedding Y ֒ → X (namely , when X is a tigh t extension of Y ). The analogue of this question for close d subspaces is somewhat unsatisfactory . W e saw that uni- form retracts, clop en subspaces, as w ell as direct summands, of straigh t spaces are alwa ys straigh t ([BDP2]). On the other hand, closed subspaces ev en of ULC spaces ma y fail to b e straight (see [BDP2]). Another instance when a closed subspace of a straigh t space fails to b e s traigh t is giv en by the f ollo wing fact prov ed in [BDP4]: the spaces X in whic h every closed subspace is straigh t are precisely the UC spaces [BDP4]. Hence ev ery straigh t space that is not UC has closed non-straigh t subspaces. This motiv ates the follo wing general Problem 2. Find a sufficien t condition ensuring that a closed subspace Y of a straigh t s pace X is s traig h t. Question 5. Gener alize the r esults on s tr aight sp ac es f r om the c ate gory of metric sp ac es to the c ate gory of uniform sp ac es. The results from §6.3 describ e when infinite pro ducts of ULC spaces are again ULC or s traigh t. The case of precompact spaces is still op en, so w e start with the f ollo wing still unsolve d 32 D. Dikranjan and D. Rep o v ˇ s Question 6. L et X b e a pr e c omp act str aight sp ac e. Is the infinite p ower X ω ne c essarily str aight? More generally: Question 7. L et X n b e a pr e c omp act str aight s p ac e f or every n ∈ N . Is the in fi nite pr o duct Q n X n ne c essarily str aight? It is easy to see that a p ositiv e answ er to this question is equiv a- len t to a p ositiv e answ er to item (b) of the f ollo wing general q ues- tion: (i.e ., the version of Theorem 30 for pro ducts of pr e c omp act spaces): Question 8. L et the m etric s p ac e Y i b e a tight extension of X i for e ach i ∈ N . (a) Is Π i Y i a tight extension of Π i X i . (b) What ab out precompact metric sp ac es Y i ? A ckno wledgements This researc h was supp orted by the gran t MTM2009-14409-C02- 01 and Slo v enian R esearc h Agency Grants P1-0292-0101, J1-2057- 0101 and J1-9643-010 1. Reference s [A1] M. Atsuji , Uniform c ontinuity of c ontinuous functions on metric sp ac es , P acif. J. Math. 8 (1958), 11-16. [A2] M. 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